Figure 1: The inverse of migration rates vs. eccentricity for four planet masses. Top left panel - 1 , top right panel - 5 , lower left panel - 10 , lower right panel -15 . Three resolutions were considered: (150, 300) (dashed line), (300, 600) (dotted line) and (450, 900) (solid line). Also shown are migration rates predicted by Eq. (12), but multiplied by a factor of three (dot-dashed line). | |
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Figure 2: Eccentricity damping rates vs. initial eccentricity. Protoplanet mass and grid resolution are the same as Fig. 1; Papaloizou & Larwood's prescription for e-damping is given as exact (i.e. no factor of 3 modification). Agreement is strong for e_{p}> 0.07. | |
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Figure 3: Migration time (computed from Eq. (11)) vs. softening for a body of 5 on a fixed circular orbit. Four torque cutoffs are represented: (solid), (dotted), 1.0 (dashed) & (dot-dashed); the latter two overlap. Also plotted is Papaloizou & Larwood's prescription for migration rates multiplied by a factor of 3 (triple dot-dashed); Tanaka et al.'s migration rates in 2D & 3D discs (lower and upper horizontal long-dashed lines); Korycansky & Pollack's results with softening parameter = 10^{-4} (shaded triangle). The peak at arises when , such that the softening length coincides with a radial cell boundary. | |
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Figure 4: Evolution of a 10 planet N-body model with the fiducial setup (see main text). Top: semi-major axes of the migrating embryos. A brief period of activity is followed by a long period of migration between bodies in first-order mean-motion resonances. Bottom: the embryos' eccentricities over the same time. Damping from the disc prevents eccentricities from growing outside periods of intense activity. | |
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Figure 5: First order resonances between five of the bodies in Fig. 4. Each plot is labelled (P_{1}, P_{2})-p:q, where P_{1}/P_{2} are the exterior/interior protoplanets respectively, and p:q gives the commensurability between them; left/right columns show the resonant argument associated with the exterior/interior body, respectively. Each plot takes the same colour as the corresponding exterior protoplanet in that resonance. Note that the y-axis differs by in the left and right columns. | |
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Figure 6: As Fig. 4, but for a model with a higher initial eccentricity distribution ( compared to previously). Resonant migration dominates again once the excited body has been scattered beyond the system of protoplanets. The two plots to the right detail the evolution of the main population. | |
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Figure 7: As Fig. 4, but with initial separations of 4 between the protoplanets. Resonant migration again dominates, but now with two pairs of co-orbital protoplanets. These two pairs lie in a 3:2 mean motion resonance, a consequence of the 14 body displacing the inner pair at yr without breaking the commensurability. The bottom plot shows a detailed view of the active period following intial settling. | |
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Figure 8: As Fig. 4, but with a Gaussian mass distribution for the protoplanets. From the object at smallest initial radius, to the nearest 0.1 the masses of the bodies are 13.8, 9.8, 17.6, 14.2, 8.5, 19.4, 2.9, 10.6, 14.7 & 6.6 . The non-monotonic distribution of migration rates causes the population to fragment into smaller resonant groups. The 2.9 body (orange) is excited rather than captured by the neighbouring 10.6 embryo (purple), but is subsequently able to form a stable resonance with another body of comparable mass. | |
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Figure 9: As Fig. 4, but for a model with 20 protoplanets separated initially by 5 . The protoplanets have reduced masses 1 , 1.5 , 2 ,..., 10.5 , reflecting an earlier period of growth. The middle and bottom plots show the eccentricities of the initially inner and outer ten bodies, respectively. Collisions increase protoplanet masses and cause the formation of several smaller resonantly migrating groups. | |
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Figure 10: As Fig. 4, but for a model with reduced (gaseous) disc mass. Eccentricity damping (and migration rates) are reduced by a factor of 5, but collisions reduce the total number of bodies while eccentricity damping is still too strong for mutual interactions to excite the population. | |
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Figure 11: As Fig. 4, but for a model with eccentricity damping (Eq. (17)) reduced by a factor of 50. Scattering of only ten bodies is now self-sustaining over yr, redistributing bodies throughout the disc. Without energy losses to the disc, migration is reversed at even moderate eccentricities. | |
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Figure 12: Evolution of a model of ten bodies using the NIRVANA hydrodynamic code, similar to Fig. 4. Collisions remove smaller, easily perturbed bodies as differential migration produces a successively more crowded system, until stable resonances form and the group migrate inwards at a single rate. As in the majority of N-body integrations, no body achieves and sustains e_{p}>0.1. | |
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Figure 13: As Fig. 12, but with higher initial eccentricities, similar to Fig. 6. All surviving bodies end the run in 5:4 and/or 6:5 resonances with adjacent protoplanets, while a co-orbital system forms and remains stable until the end of integrations, over yr later. Rapid eccentricity damping is clearly visible early on in the 4 (light blue), 6 (olive green) and 2 (black) bodies; only the former acheives e_{p} > 0.1. | |
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Figure 14: First order resonances between the protoplanets in Fig. 13. Each plot is labelled and coloured as in Fig. 5. Note that the trojan planets (P_{2}=2,4) both share the two resonances with exterior bodies (P_{1}=5,6), as shown in the lower four pairs of plots. | |
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Figure 15: As Fig. 12, but with reduced initial separations , similar to Fig. 7. The excitation and subsequent collision of the smallest body is repeated from some N-body runs (cf. Fig. 4), and a co-orbital system forms for yr. Long-term stable resonances are predominantly 5:4 and 6:5. More massive bodies progress inwards by displacing smaller protoplanets to larger radii. | |
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Figure 16: As Fig. 15, but with the increased initial eccentricities of Fig. 6 ( ). Compare this and Fig. 15 with Figs. 12 and 13; in both pairs of models, the run with higher intial eccentricities does little to raise the number of encounters beyond the first 3000 years. | |
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Figure 17: As Fig. 15, but with with a randomised, Gaussian protoplanet mass distribution, similar to Fig. 8. Working radially outwards through the initial distribution, to the nearest 0.1 the bodies have masses 9.5, 14.1, 8.8, 5.4, 4.9, 11.2, 2.4, 5.9, 18.0 and 10.7 . As in the N-body case, the non-sequential arrangement of masses causes the population to split into several resonant groups, while scattering only occurs between bodies with mass ratios greater than . | |
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Figure 18: A series of surface density plots of the simulation shown in Fig. 12. The time in years of each snapshot is shown above each plot. The protoplanets are represented by white circles, with size proportional to mass. | |
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